# Show that $G$ has four inequivalent 1-dimensional complex representations

I am looking at the exercise below in relation to representation theory

We define a group structure on $$\lbrace \pm 1, \pm a, \pm b , \pm c \rbrace$$ through the relations $$a^2 =b^2 =c^2 =abc = 1$$ and denote the resulting group by G.

1. Show that $$G$$ has four inequivalent 1-dimensional complex representations, and find those (Hint: $$\lbrace \pm 1 \rbrace$$ is a normal subgroup of G and $$G / \lbrace \pm 1 \rbrace \simeq \mathbb{Z} / 2 \mathbb{Z} \times \mathbb{Z} / 2 \mathbb{Z}$$.)
2. Define $$\pi: G \to GL_2(\mathbb{C})$$ by $$\pi (\pm 1 )= \pm I_2 , \pi(\pm a)=\pm \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}, \pi (\pm b)=\pm \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \pi(\pm c)= \pm \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$$. Prove that $$\pi$$ is a well-defined irreducible representation of $$G$$.
3. Use (1) and (2) to find the character table of G.

I have shown part 2. by using the inner product of the character, but I am having some difficulties in part 1. (and therefore also in part 3). In part 1, I am not quite sure how I am to use a normal subgroup in order to show the wanted. Additionally, I am having some difficulties in deciding what the representations explicitly are.

• The answer here might be useful to you.
– Sam
Mar 12, 2021 at 15:47

The cyclic group $$\mathbb{Z}/n\mathbb{Z}$$ has 1D representations $$x\mapsto\zeta^x$$ where $$\zeta=\exp(2\pi i/n)$$ is a primitive $$n$$th root of unity. In general, a finite abelian group has only 1D irreducible representations. If $$A$$ is abelian and $$K$$ the kernel of a representation $$\rho:A\to GL_1(\mathbb{C})$$, then the image is a cyclic subgroup of $$S^1$$ (roots of unity), and by the first isomorphism theorem, $$K/A$$ must be cyclic.
Given a normal subgroup $$N\trianglelefteq G$$ and a representation $$\rho:G/N\to GL(U)$$ of its quotient, there is also a projection map $$\pi:G\to G/N$$, and the composition $$(\rho\circ\pi):G\to GL(U)$$ is a rep of $$G$$.
In particular, for $$V=\mathbb{Z}_2\times\mathbb{Z}_2$$, there are three (proper nontrivial) subgroups $$W$$, each isomorphic to $$\mathbb{Z}_2$$ generated by a nontrivial element of $$V$$. Each of the quotients $$V/W$$ has a unique nontrivial representation, taking the coset $$W$$ to $$+1$$ and the other coset of $$W$$ to $$-1$$. The corresponding representation of $$V$$ produces $$\pm1$$ depending on if an element is in $$W$$ or not (this only works because $$W$$ is index $$2$$).
Similarly, 1D reps of $$Q_8/C_2=V$$ become 1D reps of $$Q_8$$. The three aforementioned subgroups of $$V$$ become the three subgroups $$H=\langle\mathbf{i}\rangle,\langle\mathbf{j}\rangle,\langle\mathbf{k}\rangle$$ of $$Q_8$$, each index $$2$$. Then we also have the three 1D reps of $$Q_8$$ defined as producing $$\pm1$$ (within $$GL_1(\mathbb{C})$$) based on whether the element of $$Q_8$$ is in $$H$$ or not. (And of course the trivial rep is 1D.)